Frequency Methods in Oscillation Theory

Inhaltsverzeichnis

• 1. Classical two-dimensional oscillating systems and their multidimensional analogues.
• §1.1. The van der Pol equation.
• §1.2. The equation of oscillations of a pendulum.
• §1.3. Oscillations in two-dimensional systems with hysteresis.
• §1.4. Lower estimates of the number of cycles of a two-dimensional system.
• 2. Frequency criteria for stability and properties of solutions of special matrix inequalities.
• §2.1. Frequency criteria for stability and dichotomy.
• §2.2. Theorems on solvability and properties of special matrix inequalities.
• 3. Multidimensional analogues of the van der Pol equation.
• §3.1. Dissipative systems. Frequency criteria for dissipativity.
• §3.2. Second-order systems. Frequency realization of the annulus principle.
• §3.3. Third-order systems. The torus principle.
• §3.4. The main ideas of applying frequency methods for multidimensional systems.
• §3.5. The criterion for the existence of a periodic solution in a system with tachometric feedback.
• §3.6. The method of transition into the „space of derivatives“.
• §3.7. A positively invariant torus and the function „quadratic form plus integral of nonlinearity“.
• §3.8. The generalized Poincaré–Bendixson principle.
• §3.9. A frequency realization of the generalized Poincaré-Bendixson principle.
• §3.10. Frequency estimates of the period of a cycle.
• 4. Yakubovich auto–oscillation.
• §4.1. Frequency criteria for oscillation of systems with one differentiable nonlinearity.
• §4.2. Examples of oscillatory systems.
• 5. Cycles in systems with cylindrical phase space.
• §5.1. The simplest case of application of the nonlocal reduction method for the equation of a synchronous machine.
• §5.2. Circular motions and cycles of the second kind in systems with one nonlinearity.
• §5.3. The method ofsystems of comparison.
• §5.4. Examples.
• §5.5. Frequency criteria for the existence of cycles of the second kind in systems with several nonlinearities.
• §5.6. Estimation of the period of cycles of the second kind.
• 6. The Barbashin-Ezeilo problem.
• §6.1. The existence of cycles of the second kind.
• §6.2. Bakaev stability. The method of invariant conical grids.
• §6.3. The existence of cycles of the first kind in phase systems.
• §6.4. A criterion for the existence of nontrivial periodic solutions of a third-order nonlinear system.
• 7. Oscillations in systems satisfying generalized Routh-Hurwitz conditions. Aizerman conjecture.
• §7.1. The existence of periodic solutions of systems with nonlinearity from a Hurwitzian sector.
• §7.2. Necessary conditions for global stability in the critical case of two zero roots.
• §7.3. Lemmas on estimates of solutions in the critical case of one zero root.
• §7.4. Necessary conditions for absolute stability of nonautonomous systems.
• §7.5. The existence of oscillatory and periodic solutions of systems with hysteretic nonlinearities.
• 8. Frequency estimates of the Hausdorff dimension of attractors and orbital stability of cycles.
• §8.1. Upper estimates of the Hausdorff measure of compact sets under differentiable mappings.
• §8.2. Estimate of the Hausdorff dimension of attractors of systems of differential equations.
• §8.3. Global asymptotic stability of autonomous systems.
• §8.4. Zhukovsky stability of trajectories.
• §8.5. A frequency criterion for Poincaré stability of cycles of the second kind.
• §8.6. Frequency estimates for the Hausdorff dimension and conditions for global asymptotic stability.